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I am reading Dummit and Foote, Chapter 13 - Field Theory.

I am currently studying Section 13.2 : Algebraic Extensions

I need some help with an aspect of Propositions 11 and 12 ... ...

Propositions 11 and 12 read as follows:

Now Proposition 11 states that the degree of ##F( \alpha )## over ##F## is

##[ F( \alpha ) \ : \ F ] = \text{ deg } m_\alpha (x) = \text{ deg } \alpha##

... ... BUT ... ...

... ... Proposition 12 states that ... "if ##\alpha## is an element of an extension of degree ##n## over ##F##, then ##\alpha## satisfies a polynomial of degree

Doesn't Proposition 11 guarantee that the polynomial (the minimum polynomial) is actually of degree equal to ##n##???

Can someone please explain in simple terms how these statements are consistent?

Help will be appreciated ...

Peter

I am currently studying Section 13.2 : Algebraic Extensions

I need some help with an aspect of Propositions 11 and 12 ... ...

Propositions 11 and 12 read as follows:

Now Proposition 11 states that the degree of ##F( \alpha )## over ##F## is

__the degree of the minimum polynomial ... ... that is__**equal to**##[ F( \alpha ) \ : \ F ] = \text{ deg } m_\alpha (x) = \text{ deg } \alpha##

... ... BUT ... ...

... ... Proposition 12 states that ... "if ##\alpha## is an element of an extension of degree ##n## over ##F##, then ##\alpha## satisfies a polynomial of degree

__... ... "__**at most ##n## over ##F##**Doesn't Proposition 11 guarantee that the polynomial (the minimum polynomial) is actually of degree equal to ##n##???

Can someone please explain in simple terms how these statements are consistent?

Help will be appreciated ...

Peter